Without loss of generality, in this paper, we focus on soft clustering which makes it easier to define a loss using the probabilities and update the parameters using the gradients to enable end-to-end learning. We follow the soft clustering framework presented in SwAV [1]}, which is a centroid based technique and aims to maintain consistency between the clusterings of the augmented views \(\mathcal {X}_b^{1}\)  and \(\mathcal {X}_b^{2}\) . We store a set of randomly initialized prototypes \(C_0=\lbrace  \mathbf {c}_0^1,\cdots ,\mathbf {c}_0^K \rbrace  \in \mathbb {R}^{d\times K}\) , where \(K\)  is the number of clusters, and \(d\)  is the dimension of the prototypes. These prototypes are used to represent clusters and define a “consensus consistency" loss. We compute \(M\)  variations of \(C_0\)  as \(C_1,...,C_M\)  exactly like we compute \(M\)  variation of \(\mathcal {Z}_0\) .
